How not to do Instrumental variables

Instrumental Variable estimation, Generalized Method of Moments and related techniques are part of the standard toolkit for applied economists. They are also increasingly used in other fields such as health.
What everyone knows, or should know, is that while one can think of these models as a two stage process this is not actually how you do it. But this paper which looks at how systolic blood pressure depends on anti-hypertensive drugs in Japan, published in the Bulletin of the World Health Organization 2008, gets it badly wrong. As they note, a simple regression of blood pressure on medication is likely to get a positive slope so you need to instrument or do something.
They estimate a logit and then stick the predicted values into an OLS model. Aside from the identifying assumption (which isn't discussed & looks pretty dodgy to me), this is not IV as usually defined and it is not clear that the estimate is consistent or that the standard errors are correct. The model also includes controls for exercise but these are also likely to be endogenous but this is ignored.